ABSTRACT We consider the following problem: can a certain graph parameter of some given graph be reduced by at most $d$ for some integer~$d$ via at most $k$ edge contractions for some given integer $k$? We consider three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when $d$ is part of the input, this problem is polynomial-time solvable on $P_4$-free graphs and \NP-complete for split graphs. Moreover, as split graphs form a subclass of $P_5$-free graphs, both results together give a complete complexity classification for $P_\ell$-free graphs. On the positive side, for split graphs, we also show that the problem is polynomial-time solvable if $d$ is fixed, that is, not part of the input.
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